ranking systems

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The Page Rank Axioms Based on Ranking Systems: The PageRank Axioms, by Alon Altman and Moshe Tennenholtz. Presented by Aron Matskin

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Page 1: Ranking systems

The Page Rank Axioms

Based on Ranking Systems: The PageRank Axioms, by Alon Altman

and Moshe Tennenholtz.Presented by Aron Matskin

Page 2: Ranking systems

Judge and be prepared to be judged.Ayn Rand

רבי שמעון אומר שלשה כתרים הם: כתר תורה, וכתר כהונה, וכתר מלכות; וכתר שם

טוב עולה על גביהן.

פירקי אבות

Page 3: Ranking systems

Talking Points

Ranking and reputation in general Connections to the Internet world PageRank web ranking system PageRank representation theorem

Page 4: Ranking systems

Ranking: What

Abilities Choices Reputation Quality

Quality of information

Popularity Good looks What not?

Page 5: Ranking systems

Ranking: How

Voting Reputation systems Peer review Performance reviews Sporting competition Intuitive or ad-hoc

Page 6: Ranking systems

Ranking Systems’ Properties

Ad-hoc or systematic Centralized or distributed Feedback or indicator-based Peer, “second-party”, or third-party Update period Volatility Other?

Page 7: Ranking systems

Agents Ranking Themselves

Community reputation Professional associations Peer review Performance reviews (in part) Web page ranking

Page 8: Ranking systems

Ranking: Problems and Issues Eliciting information Information aggregation Information distribution Truthfulness

Strategic considerations Fear of retribution / expectation of kick-backs Coalition formation

Agent identification (pseudonym problem)

Need analysis!

Page 9: Ranking systems

Ranking Systems: Analysis

Empirical Because theories often lack

Theoretical Because theoreticians need to eat, too Provides valuable insight

Page 10: Ranking systems

Social Choice Theory

Two approaches: Normative – from properties to

implementations. Example: Arrow’s Impossibility Theorem

Descriptive – from implementation to properties. The Holy Grail: representation theorems (uniqueness results)

Page 11: Ranking systems

PageRank Method

A method for computing a popularity (or importance) ranking for every web page based on the graph of the web.

Has applications in search, browsing, and traffic estimation.

Page 12: Ranking systems

PageRank: Intuition

Internet pages form a directed graph

Node’s popularity measure is a positive real number. The higher number represents higher popularity. Let’s call it weight

Node’s weight is distributed equally among nodes it links to

We look for a stationary solution: the sum of weights a page receives from its backlinks is equal to its weight

b=2

c=1

a=2

1

1

1

1

Page 13: Ranking systems

PageRank as Random Walk

Suppose you land on a random page and proceed by clicking on hyper-links uniformly randomly

Then the (normalized) rank of a page is the probability of visiting it

Page 14: Ranking systems

PageRank: Some Math

Represent the graph as a matrix:

b

c

a 010

½01

½00

a b c

a

b

c

Page 15: Ranking systems

PageRank: Some Math

Find a solution of the equation:

AG r = r

Under the assumption that the graph is strongly connected there is only one normalized solution The assumption is not used by the real PageRank algorithm which uses workarounds to overcome it

The solution r is the rank vector.

Page 16: Ranking systems

Calculating PageRank

Take any non-zero vector r0

Let ri+1 = AG ri

Then the sequence rk converges to r

Since the Internet graph is an expander, the convergence is very fast: O(log n) steps to reach given precision

Page 17: Ranking systems

PageRank: The Good News

Intuitive Relatively easy to calculate Hard to manipulate Great for common case searches May be used to assess quality of

information (assuming popularity ≈ trust)

Page 18: Ranking systems

PageRank: The Bad News

PageRank is proprietary to Webmasters can’t manipulate it,

but can Every change in the algorithm is good

for someone and is bad for someone else

Popular become more popular Popularity ≠ quality of information

Page 19: Ranking systems

The Representation Theorem We next present a set of axioms (i.e.

properties) for ranking procedures Some of the axioms are more intuitive

then others, but all are satisfied by PageRank

We then show that PageRank is the only ranking algorithm that satisfies the axioms

We try to be informal, but convincing

Page 20: Ranking systems

Ranking Systems Defined

A ranking system F is a functional that maps every finite strongly connected directed graph (SCDG) G=(V,E) into a reflexive, transitive, complete, and anti-symmetric binary relation ≤ on V

Page 21: Ranking systems

Ranking Systems: Example MyRank ranks vertices in G in ascending

order of the number of incoming links

b

c

aMyRank(G): c = a < b

PageRank(G): c < a = b

Page 22: Ranking systems

Axiom 1: Isomorphism (ISO)

F satisfies ISO iff it is independent of vertex names Consequence: symmetric vertices

have the same rank

b

e

a

gf

j

i

he = f = g = h = i = j

a = b

Page 23: Ranking systems

Axiom 2: Self Edge (SE) Node v has a self-edge (v,v) in G’, but

does not in G. Otherwise G and G’ are identical. F satisfies SE iff for all u,w ≠ v:(u ≤ v u <’ v) and (u ≤ w u ≤’ w)

PageRank satisfies SE:Suppose v has k outgoing edges in G. Let (r1,…,rv,…,rN) be the rank vector of G, then (r1,…,rv+1/k,…,rN) is the rank vector of G’

Page 24: Ranking systems

Axiom 3: Vote by Committee (VBC)

a

c

b

a

c

b

1. In the example page a links only to b and c, but there may be more successors of a

2. Incoming links of a and all other links of the successors of a remain the same

Page 25: Ranking systems

Axiom 4: Collapsing (COL)

b

a

b

1. The sets of predecessors of a and b are disjoint

2. Pages a and b must not link to each other or have self-links

3. The sets of successors of a and b coincide

Page 26: Ranking systems

Axiom 5: Proxy (PRO)

1. All predecessors of x have the same rank2. |P(x)| = |S(x)|3. x is the only successor of each of its

predecessors

x

=

=

Page 27: Ranking systems

Useful Properties: DEL

1. |P(b)|=|S(b)|=12. There is no direct edge between a and c3. a and c are otherwise unrestricted

a

cb

d

a

c

d

Page 28: Ranking systems

DEL: Proof

a

cb

d

cb

d

a

VBC

Page 29: Ranking systems

DEL: Proof

cb

d

a

VBCcb

d

a

Page 30: Ranking systems

DEL: Proof

ISO,PROcb

d

a

cb

d

a

Page 31: Ranking systems

DEL: Proof

PROc

d

a

cb

d

a

Page 32: Ranking systems

DEL: Proof

PROc

d

a

c

d

a

Page 33: Ranking systems

DEL: Proof

VBCc

d

a

c

d

a

Page 34: Ranking systems

DEL: Proof

VBCc

d

a a

c

d

Page 35: Ranking systems

DEL for Self-Edge

It can also be shown that DEL holds for self-edges:

a a

Page 36: Ranking systems

Useful Properties: DELETE

1. Nodes in P(x) have no other outgoing edges

2. x has no other edges

x

=

=

=

=

Page 37: Ranking systems

DELETE: Proof

x

=

=

=

=

COL

x

y

Page 38: Ranking systems

DELETE: Proof

PRO

x

y

Page 39: Ranking systems

Useful Properties: DUPLICATE

1. All successors of a are duplicated the same number of times

2. There are no edges from S(a) to S(a)

c

b

d

a c

b

d

a

Page 40: Ranking systems

DUPLICATE: Proof

c

b

d

a c

b

d

a

VBC

Page 41: Ranking systems

DUPLICATE: Proof

c

b

d

a

VBC

c

b

d

a

Page 42: Ranking systems

DUPLICATE: Proof

c

b

d

a

COL

c

b

d

a

Page 43: Ranking systems

DUPLICATE: Proof

c

b

d

a

ISO,PRO

c

b

d

a

Page 44: Ranking systems

DUPLICATE: Proof

c

b

d

a

COL-1

c

b

d

a

Page 45: Ranking systems

DUPLICATE: Proof

VBC-1

c

b

d

a c

b

d

a

Page 46: Ranking systems

The Representation Theorem Proof

Given a SCDG G=(V,E) and a,b in V, we eliminate all other nodes in G while preserving the relative ranking of a and b

In the resulting graph G’ the relative ranking of a and b given by the axioms can be uniquely determined. Therefore the axioms rank any SCDG uniquely

It follows that all ranking systems satisfying the axioms coincide

Page 47: Ranking systems

Proof by Example on b and d

b

c

a

a b c

a

b

cd

⅓00½

⅓00½

⅓000

0110d

d

3

3

1

4

a

b

c

d

Page 48: Ranking systems

Step 1: Insert Nodes

b

c

a

d

b

c

a

d

By DEL the relative ranking is preserved

Page 49: Ranking systems

Step 2: Choose Node to Remove

b

c

a

d

Page 50: Ranking systems

Step 3: Remove “self-edges”

b

c

a

d

Page 51: Ranking systems

Step 4: Duplicate Predecessors

b

c

a

d

Page 52: Ranking systems

Step 5: DELETE the Node

b

cd

Page 53: Ranking systems

Step 5: DELETE the Extras

There still are nodes to delete: back to Step 2

b

cd

Page 54: Ranking systems

Step 2: Choose Node to Remove

Steps 3,4 - no changes

b

cd

Page 55: Ranking systems

Step 5: DELETE the Node

b

d

Page 56: Ranking systems

Step 6: DELETE the Extras

No original nodes to remove: proceed to Step 7

b

d

Page 57: Ranking systems

Step 7: Balance by Duplication

b

d

This is our G’

Page 58: Ranking systems

Step 8: Equalize by Reverse DEL

b

dBy ISO b=d. By DEL and SE: in G’ b<d.

Page 59: Ranking systems

Example for a and d

b

c

a

d

b

c

a

d

Page 60: Ranking systems

After Removal of c

ba

d

Page 61: Ranking systems

Duplicate Predecessors of b

ba

d

Page 62: Ranking systems

DELETE b

a

d

Page 63: Ranking systems

DELETE Extras

a

d

Page 64: Ranking systems

Before Balancing

a

d

Page 65: Ranking systems

After Balancing

a

dConclusion: a<d.

Page 66: Ranking systems

What about a and b?

ba

d

Page 67: Ranking systems

What about a and b?

ba

d

Page 68: Ranking systems

What about a and b?

ba

Page 69: Ranking systems

What about a and b?

ba

Page 70: Ranking systems

What about a and b?

ba

Page 71: Ranking systems

What about a and b?

ba

Conclusion: a=b.

Page 72: Ranking systems

Concluding Remarks

‘Representation theorems isolate the “essence” of particular ranking systems, and provide means for the evaluation (and potential comparison) of such systems’ – Alon & Tennenholtz

Page 73: Ranking systems

The Endc

b

d

a

½0

0½0

101

0a b

c

a

b

c